Problem: Two lines have the same non-zero $y$-intercept. The first line has a slope of 10 and an $x$-intercept of $(s, 0)$. The second line has a slope of 6 and an $x$-intercept of $(t, 0)$. What is the ratio of $s $ to $t$? Express your answer as a common fraction.
Answer: The equation of the first line is $y = 10 x + b$ where $b$ is the $y$-intercept of the two lines.  Since $(s, 0)$ lies on the line, we can plug this into the line's equation to get $0 = 10s + b\Rightarrow s = -\frac b{10}$.  Similarly, the second line has equation  $y = 6 x + b$.  Plugging $(t, 0)$ into this equation gives $0 = 6t + b \Rightarrow t = - \frac b6$.  Thus $\frac st = -\frac b{10} \cdot - \frac 6b = \boxed{\frac 35}$.